The olive oil picks up extra flavor from the golden garlic in this dish. Slideshow: More Spaghetti Recipes
In a large heavy skillet, heat the oil over medium heat until hot, then stir in the garlic and cook, stirring, until golden, about 2 minutes. Using a slotted spoon, transfer the garlic to paper towels, reserving the oil in the skillet.
Toss the chicken with 1/2 teaspoon each salt and pepper, then add it to the oil in the skillet. Cook the chicken, stirring occasionally, until golden and cooked, about 8 minutes. Stir in the broccoli and pepper flakes and cook until hot, about 3 minutes. Season the chicken mixture with salt and pepper to taste.
In a large pot of boiling salted water, cook the spaghetti until al dente. Reserve 1/2 cup of the pasta cooking water, then drain the pasta. Toss the pasta with the chicken mixture in the skillet along with 1/4 cup of the pasta cooking water. Add more pasta cooking water for a thinner sauce, then season with salt and pepper to taste and serve sprinkled with the reserved garlic.